Questions S1 - A-Level Maths

$\boldsymbol { x }$	20	25	30	35	40	45	50	55	60	65	70	75
$\boldsymbol { y }$	242	262	269	290	298	310	326	355	359	375	390	412

Calculate the equation of the least squares regression line of $y$ on $x$.
Interpret, in context, your value for the gradient of this regression line.
Use your equation to estimate the mass of the compound which will dissolve in 100 ml of water at $68 ^ { \circ } \mathrm { C }$.
Given that the values of the 12 residuals for the regression line of $y$ on $x$ lie between - 7 and + 9 , comment, with justification, on the likely accuracy of your estimate in part (c).
[0pt] [2 marks]

AQA S1 2016 June Q5

7 marks

5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, $X$ millilitres, in a bottle may be modelled by a normal distribution with mean $\mu = 1525$ and standard deviation $\sigma = 9.6$.

Determine the probability that the volume of water in a randomly selected bottle is:
1. less than 1540 ml ;
2. more than 1535 ml ;
3. between 1515 ml and 1540 ml ;
4. not 1500 ml .
The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of $\sigma$, calculate the reduction in the value of $\mu$ in order to satisfy this requirement. Give your answer to one decimal place.
Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
  [0pt] [7 marks]

AQA S1 2016 June Q6

2 marks

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

Colour	Blue	Green	Red	Orange	Yellow	White
Proportion	0.25	0.25	0.18	0.12	0.15	0.05

A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with $n = 50$, together with relevant information from the table, to estimate the probability that this sample contains:
1. exactly 4 red loom bands;
2. at most 10 yellow loom bands;
3. at least 30 blue or green loom bands;
4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
The random variable $R$ denotes the number of red loom bands in a random sample of $\mathbf { 3 0 0 }$ loom bands selected from the box. Estimate values for the mean and the variance of $R$.
[0pt] [2 marks]

AQA S1 2016 June Q7

5 marks

7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( $\pounds$ ). The amounts paid, $\pounds x$, by a sample of 40 customers were, in ascending order, as follows.

Edexcel S1 Q1

A histogram is to be drawn to represent the following grouped continuous data:

Group	$0 - 10$	$10 - 20$	$20 - 25$	$25 - 30$	$30 - 50$	$50 - 100$
Frequency	$2 x$	$3 x$	$5 x$	$6 x$	$2 x$	$x$

The ' $10 - 20$ ' bar has height 6 cm and width 4 cm . Calculate

the height of the ' $20 - 25$ ' bar,
the total area under the histogram.

Edexcel S1 Q2

2. The events $A$ and $B$ are independent. Given that $\mathrm { P } ( A ) = 0.4$ and $\mathrm { P } ( A \cap B ) = 0.12$, find

$\mathrm { P } ( B )$,
$\mathrm { P } ( A \cup B )$,
$\mathrm { P } \left( A ^ { \prime } \cap B \right)$,
$\mathrm { P } \left( A \mid B ^ { \prime } \right)$.

Edexcel S1 Q3

3. The random variable $X$ has the discrete uniform distribution over the set of consecutive integers $\{ - 7 , - 6 , \ldots , 10 \}$.
Calculate (a) the expectation and variance of $X$,
(b) $\mathrm { P } ( X > 7 )$,
(c) the value of $n$ for which $\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }$.

Edexcel S1 Q4

4. The marks, $x$ out of 100 , scored by 30 candidates in an examination were as follows:

5	19	20	21	23	25	31	37	39	41
42	44	47	51	56	57	60	61	62	65
67	70	71	73	75	77	81	82	98	100

Given that $\sum x = 1600$ and $\sum x ^ { 2 } = 102400$,

find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, $y$, using the formula $y = \frac { 4 x } { 5 } + 20$.
Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}

Edexcel S1 Q5

The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.

Registration letter	$K$	$L$	$M$	$N$	$P$	$R$	$S$	$T$	$V$
Number of cars $( x )$	6	7	9	11	15	14	12	10	7
Number of vans $( y )$	8	10	14	13	13	15	14	9	8

Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
If there were $4 J$-registered cars, estimate the number of $J$-registered vans. Given that $\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264$ and $\sum x y = 1106$,
calculate the product-moment correlation coefficient between $x$ and $y$. Give a brief interpretation of your answer.

Edexcel S1 Q6

6. The distributions of two independent discrete random variables $X$ and $Y$ are given in the tables:

$x$	0	1	2
$\mathrm { P } ( X = x )$	$\frac { 3 } { 5 }$	$\frac { 3 } { 10 }$	$\frac { 1 } { 10 }$

$y$	0	1
$\mathrm { P } ( Y = y )$	$\frac { 5 } { 8 }$	$\frac { 3 } { 8 }$

The random variable $Z$ is defined to be the sum of one observation from $X$ and one from $Y$.

Tabulate the probability distribution for $Z$.
Calculate $\mathrm { E } ( Z )$.
Calculate (i) $\mathrm { E } \left( Z ^ { 2 } \right)$, (ii) $\operatorname { Var } ( Z )$.
Calculate Var (3Z-4).

Edexcel S1 Q7

7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean $5 \cdot 2$ hours. It is found that $62 \cdot 5 \%$ of the people took more than $4 \cdot 5$ hours to read the book.

Show that the standard deviation of the times is approximately $2 \cdot 2$ hours.
Calculate the percentage of the people who took between 4 and 7 hours to read the book.
Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
If a number of extra people were taken into account, all of whom took exactly $5 \cdot 2$ hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.

Edexcel S1 Q1

(a) Explain briefly what is meant by a discrete random variable.

A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable $X$ represents the number of dogs chosen.
(b) Copy and complete the table to show the probability distribution of $X$ :

$x$	0	1	2
$\mathrm { P } ( X = x )$

$\mathrm { E } ( X )$,
$\operatorname { Var } ( X )$,
$\operatorname { Var } ( 2 X )$.

Edexcel S1 Q2

2. The discrete random variable $X$ can take any value in the set $\{ 1,2,3,4,5,6,7,8 \}$. Arthur, Beatrice and Chris each carry out trials to investigate the distribution of $X$. Arthur finds that $\mathrm { P } ( X = 1 ) = 0.125$ and that $\mathrm { E } ( X ) = 4.5$.
Beatrice finds that $\mathrm { P } ( X = 2 ) = \mathrm { P } ( X = 3 ) = \mathrm { P } ( X = 4 ) = p$.
Chris finds that the values of $X$ greater than 4 are all equally likely, with each having probability $q$.

Calculate the values of $p$ and $q$.
Give the name for the distribution of $X$.
Calculate the standard deviation of $X$.

Edexcel S1 Q3

3. The marks obtained by ten students in a Geography test and a History test were as follows:

Student	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Geography $( x )$	34	57	49	21	84	53	10	77	61	85
History $( y )$	40	49	55	40		71	39	47	65	73

Given that $\sum y = 547$, calculate the mark obtained by student $E$ in History. Given further that $\sum x ^ { 2 } = 34087 , \sum y ^ { 2 } = 31575$ and $\sum x y = 31342$, calculate
the product moment correlation coefficient between $x$ and $y$,
an equation of the regression line of $y$ on $x$,
an estimate of the History mark of student $K$, who scored 70 in Geography.
State, with a reason, whether you would expect your answer to part (d) to be reliable. \section*{STATISTICS 1 (A) TEST PAPER 2 Page 2}

Edexcel S1 Q4

The random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$.
1. If $2 \mu = 3 \sigma$, find $\mathrm { P } ( X < 2 \mu )$.
2. If, instead, $\mathrm { P } ( X < 3 \mu ) = 0 \cdot 86$,
  1. find $\mu$ in terms of $\sigma$,
  2. calculate $\mathrm { P } ( X > 0 )$.
3. The stem-and-leaf diagram shows the values taken by two variables $A$ and $B$.
$A$ $B$
$8,7,4,1,0$ 1 $1,1,2,5,6,8,9$
$9,8,7,6,6,5,2$ 2 $0,3,4,6,7,7,9$
$9,7,6,4,2,1,0$ 3 $1,4,5,5,8$
$8,6,3,2,2$ 4 $0,2,6,6,9,9$
$6,4,0$ 5 $2,3,5,7$
$5,3,1$ 6 0,1
Key : 3| 1 | 2 means $$A = 13 , B = 12$$
For each set of data, calculate estimates of the median and the quartiles.
Calculate the 42nd percentile for $A$.
On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data.
Describe the skewness of the distribution of $A$ and of $B$.

Edexcel S1 Q6

6. The values of the two variables $A$ and $B$ given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled $A$ and $B$. One card is selected from Pack A. Let $A _ { i }$ represent the event that the first digit on this card is $i$.

Write down the value of $\mathrm { P } \left( A _ { 2 } \right)$. The card taken from Pack $A$ is now transferred into Pack $B$, and one card is picked at random from Pack $B$. Let $B _ { i }$ represent the event that the first digit on this card is $i$.
Show that $\mathrm { P } \left( A _ { 1 } \cap B _ { 1 } \right) = \frac { 1 } { 24 }$.
Show that $\mathrm { P } \left( A _ { 6 } \mid B _ { 5 } \right) = \frac { 4 } { 41 }$.
Find the value of $\mathrm { P } \left( A _ { 1 } \cup B _ { 3 } \right)$.

Edexcel S1 Q1

(a) Explain briefly what is meant by a random variable.
(b) Write down a quantity which could be modelled as
1. a discrete random variable,
2. a continuous random variable.
3. The discrete random variable $X$ has the probability function given by the following table:
$x$ 0 1 2 3 4 5 6
$\mathrm { P } ( X = x )$ 0.09 0.12 0.22 0.16 $p$ $2 p$ 0.2
(a) Show that $p = 0.07$
(b) Find the value of $\mathrm { E } ( X + 2 )$.
(c) Find the value of $\operatorname { Var } ( 3 X - 1 )$.

Edexcel S1 Q3

3. Twenty pairs of observations are made of two variables $x$ and $y$, which are believed to be related. It is found that $$\sum x = 200 , \quad \sum y = 174 , \quad \sum x ^ { 2 } = 6201 , \quad \sum y ^ { 2 } = 5102 , \quad \sum x y = 5200 .$$ Find

the product-moment correlation coefficient between $x$ and $y$,
the equation of the regression line of $y$ on $x$. Given that $p = x + 30$ and $q = y + 50$,
find the equation of the regression line of $q$ on $p$, in the form $q = m p + c$.
Estimate the value of $q$ when $p = 46$, stating any assumptions you make.

Edexcel S1 Q4

4. The heights of the students at a university are assumed to follow a normal distribution. $1 \%$ of the students are over 200 cm tall and 76\% are between 165 cm and 200 cm tall. Find

the mean and the variance of the distribution,
the percentage of the students who are under 158 cm tall.
Comment briefly on the suitability of a normal distribution to model such a population. \section*{STATISTICS 1 (A) TEST PAPER 3 Page 2}

Edexcel S1 Q5

In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:

17	12	9	23	40	32	11	5	34	22	31	8
15	45	10	52	14	13	29	43	69	30	15	47
35	6	24	13	19	26	9	31	27	18	6	20
22	18	30	51	49	35	50	25	8	10	26	31
33	29	40	37	38	44	24	34	42	38	11	23

Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5-9, 10-14, 15-19, 20-24, etc.
Find the three quartiles for the distribution.
On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers.

Edexcel S1 Q6

6. Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is $\frac { 1 } { 3 }$.

Show that $r$, the number of red cards in the pack, satisfies the equation $$r ( r - 1 ) = 420$$
Hence or otherwise find the value of $r$.
Find the probability that, when three cards are drawn at random from the pack,
1. at least two are red,
2. the first one is red given that at least two are red.

Edexcel S1 Q1

Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable $X$ takes the value of the number on the card each time the experiment is repeated.
1. What must be assumed about the cards if the distribution of $X$ is modelled by a discrete uniform distribution?
2. Making this modelling assumption, find the expectation and the variance of $X$.
3. (a) Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class.
In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was $15 - 20$ (i.e. $\mathrm { f } x$, where $15000 \leq x < 20000$ ), with frequency 300 . The highest frequency in a class was 400 , for the class $30 - 40$, and the bar representing this class was 8 cm high. The total area under the histogram was $50 \mathrm {~cm} ^ { 2 }$.
Find the height and the width of the bar representing the modal class.

Edexcel S1 Q3

3. The variable $X$ represents the marks out of 150 scored by a group of students in an examination. The following ten values of $X$ were obtained: $$60,66,76,80,94,106,110,116,124,140 .$$

Write down the median, $M$, of the ten marks.
Using the coding $y = \frac { x - M } { 2 }$, and showing all your working clearly, find the mean and the standard deviation of the marks.
Find $\mathrm { E } ( 3 X - 5 )$.

Edexcel S1 Q4

4. The discrete random variable $X$ has probability function $\mathrm { P } ( X = x ) = k ( x + 4 )$. Given that $X$ can take any of the values $- 3 , - 2 , - 1,0,1,2,3,4$,

find the value of the constant $k$.
Find $\mathrm { P } ( X < 0 )$.
Show that the cumulative distribution $\mathrm { F } ( x )$ is given by $$\mathrm { F } ( x ) = \lambda ( x + 4 ) ( x + 5 )$$ where $\lambda$ is a constant to be found. \section*{STATISTICS 1 (A) TEST PAPER 4 Page 2}

Edexcel S1 Q5

The events $A$ and $B$ are such that $\mathrm { P } ( A \cap B ) = 0.24 , \mathrm { P } ( A \cup B ) = 0.88$ and $\mathrm { P } ( B ) = 0.52$.
1. Find $\mathrm { P } ( A )$.
2. Determine, with reasons, whether $A$ and $B$ are
  1. mutually exclusive,
  2. independent.
3. Find $\mathrm { P } ( B \mid A )$.
4. Find $\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)$.
5. The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours.
  Use this model to calculate
6. the probability that a person chosen at random took between 5 and 9 hours to complete the task,
7. the range, symmetrical about the mean, within which $80 \%$ of the people's times lie.
  (5 marks)
  It is found that, in fact, $80 \%$ of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
8. Find the standard deviation of the times in the modified model.
9. The following data was collected for seven cars, showing their engine size, $x$ litres, and their fuel consumption, $y \mathrm {~km}$ per litre, on a long journey.
Car $A$ $B$ $C$ $D$ $E$ $F$ $G$
$x$ 0.95 1.20 1.37 1.76 2.25 2.50 2.875
$y$ 21.3 17.2 15.5 19.1 14.7 11.4 9.0
$\sum x = 12 \cdot 905 , \sum x ^ { 2 } = 26 \cdot 8951 , \sum y = 108 \cdot 2 , \sum y ^ { 2 } = 1781 \cdot 64 , \sum x y = 183 \cdot 176$.
Calculate the equation of the regression line of $x$ on $y$, expressing your answer in the form $x = a y + b$.
Calculate the product moment correlation coefficient between $y$ and $x$ and give a brief interpretation of its value.
Use the equation of the regression line to estimate the value of $x$ when $y = 12$. State, with a reason, how accurate you would expect this estimate to be.
Comment on the use of the line to find values of $x$ as $y$ gets very small.

Group	\(0 - 10\)	\(10 - 20\)	\(20 - 25\)	\(25 - 30\)	\(30 - 50\)	\(50 - 100\)
Frequency	\(2 x\)	\(3 x\)	\(5 x\)	\(6 x\)	\(2 x\)	\(x\)

\(x\)	0	1	2
\(\mathrm { P } ( X = x )\)	\(\frac { 3 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 10 }\)

\(y\)	0	1
\(\mathrm { P } ( Y = y )\)	\(\frac { 5 } { 8 }\)	\(\frac { 3 } { 8 }\)

\(A\)		\(B\)
\(8,7,4,1,0\)	1	\(1,1,2,5,6,8,9\)
\(9,8,7,6,6,5,2\)	2	\(0,3,4,6,7,7,9\)
\(9,7,6,4,2,1,0\)	3	\(1,4,5,5,8\)
\(8,6,3,2,2\)	4	\(0,2,6,6,9,9\)
\(6,4,0\)	5	\(2,3,5,7\)
\(5,3,1\)	6	0,1

\(\boldsymbol { x }\)	20	25	30	35	40	45	50	55	60	65	70	75
\(\boldsymbol { y }\)	242	262	269	290	298	310	326	355	359	375	390	412

5	19	20	21	23	25	31	37	39	41
42	44	47	51	56	57	60	61	62	65
67	70	71	73	75	77	81	82	98	100

Registration letter	\(K\)	\(L\)	\(M\)	\(N\)	\(P\)	\(R\)	\(S\)	\(T\)	\(V\)
Number of cars \(( x )\)	6	7	9	11	15	14	12	10	7
Number of vans \(( y )\)	8	10	14	13	13	15	14	9	8

\(x\)	0	1	2	3	4	5	6
\(\mathrm { P } ( X = x )\)	0.09	0.12	0.22	0.16	\(p\)	\(2 p\)	0.2

17	12	9	23	40	32	11	5	34	22	31	8
15	45	10	52	14	13	29	43	69	30	15	47
35	6	24	13	19	26	9	31	27	18	6	20
22	18	30	51	49	35	50	25	8	10	26	31
33	29	40	37	38	44	24	34	42	38	11	23

Car	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
\(x\)	0.95	1.20	1.37	1.76	2.25	2.50	2.875
\(y\)	21.3	17.2	15.5	19.1	14.7	11.4	9.0

5	19	20	21	23	25	31	37	39	41
42	44	47	51	56	57	60	61	62	65
67	70	71	73	75	77	81	82	98	100

17	12	9	23	40	32	11	5	34	22	31	8
15	45	10	52	14	13	29	43	69	30	15	47
35	6	24	13	19	26	9	31	27	18	6	20
22	18	30	51	49	35	50	25	8	10	26	31
33	29	40	37	38	44	24	34	42	38	11	23

Questions S1 (1967 questions)

Browse by module

Browse by board

AQA S1 2016 June Q4

AQA S1 2016 June Q5

AQA S1 2016 June Q6

AQA S1 2016 June Q7

Edexcel S1 Q1

Edexcel S1 Q2

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q5

Edexcel S1 Q6

Edexcel S1 Q7

Edexcel S1 Q1

Edexcel S1 Q2

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q6

Edexcel S1 Q1

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q5

Edexcel S1 Q6

Edexcel S1 Q1

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q5

5	19	20	21	23	25	31	37	39	41
42	44	47	51	56	57	60	61	62	65
67	70	71	73	75	77	81	82	98	100

17	12	9	23	40	32	11	5	34	22	31	8
15	45	10	52	14	13	29	43	69	30	15	47
35	6	24	13	19	26	9	31	27	18	6	20
22	18	30	51	49	35	50	25	8	10	26	31
33	29	40	37	38	44	24	34	42	38	11	23